We now derive a formula which leads to equation (REF ).
Using the transformation formulas 8.152 from [1]}, p.915 (the 7-th row) – where \(k^{\prime }=\sqrt{1-k^2}\) , \(k_1=\frac{1-k^{\prime }}{1+k^{\prime }}\) , and \(u_1=(1+k^{\prime })u\)  – we find
\(\rm {dn}(u_1,k_1) + k_1 \, \rm {cn(u_1,k_1)} = \frac{1-(1-k^{\prime }) \rm {sn}^2(u,k)}{dn(u,k)} + \frac{1-k^{\prime }}{1+k^{\prime }} \, \frac{1-(1+k^{\prime }) \rm {sn}^2(u,k)}{dn(u,k)}\) 
